How do you use linear algebra and calculus to implement natural language processing techniques?

Another set of important ways of representing text — and these are by now the most widely used methods — are *learned* representations of words/phrases/paragraphs/documents such that “similar” elements are mapped to “similar” points in some vector space. (The first similar refers to whatever makes us humans consider one piece of text similar to another, whereas the second is a mathematical similarity, such as cosine similarity.) These can be learned in a variety of different ways; some useful keywords to dive deeper into are: skip-gram, continuous bag-of-words, word2vec, glove, and GPT.

Vectorization techniques include: Bag of Words (BoW): The Bag of Words approach involves representing each word in the vocabulary as a unique index and representing each document as a vector of word frequencies. This approach captures the word co-occurrence information in the text but does not consider the order of the words. Term Frequency-Inverse Document Frequency (TF-IDF): The TF-IDF approach is similar to the BoW approach but takes into account the importance of each word in the document and the corpus. This approach assigns a weight to each word in the document based on its frequency in the document and the inverse frequency of the word in the corpus.

Vectorization techniques continued: Word Embeddings: Word embeddings are dense, low-dimensional vectors that represent words in a way that captures their semantic and syntactic meaning. Word embeddings are trained on large amounts of text data using neural networks and can capture complex relationships between words. Contextualized Word Embeddings: Contextualized word embeddings are similar to word embeddings but take into account the context in which the word appears in the text. These embeddings are trained on large amounts of text data using neural networks and can capture the semantic and syntactic meaning of the word in context.

4. Term-Document Matrix: A term-document matrix is a matrix where each row represents a term or word in the vocabulary, and each column represents a document. The matrix is filled with term frequency values, indicating how many times each term appears in each document. This type of matrix is commonly used in text classification, topic modeling, and information retrieval.

3. Co-occurrence Matrix: A co-occurrence matrix is a matrix that represents the frequency of word co-occurrences in a corpus of text. Each row and column represents a word in the vocabulary, and the matrix is filled with values indicating how often two words co-occur within a certain context window. This type of matrix is commonly used in word similarity and association analysis.

An eigenvector of a matrix is a non-zero vector that, when multiplied by the matrix, results in a scalar multiple of itself. The scalar multiple is called the eigenvalue. Eigenvalues and eigenvectors are used in many NLP applications, such as principal component analysis and dimensionality reduction.

The chain rule is a formula for computing the derivative of a composite function. In NLP, the chain rule is used in backpropagation to compute the gradients of neural networks with multiple layers.

This is not just applicable to NLP. All deep learning systems rely on gradient descent (or its stochastic sibling). It is worth noting that the calculus is more often than not handled by whatever frameworks that are available, such as PyTorch and TensorFlow. Nonetheless, it is important to have a grasp of what your system is doing, and the mathematical understanding to debug it and improve its performance as necessary.

Backpropagation is a technique used to compute the gradients efficiently in deep learning models. In NLP, backpropagation is used to compute the gradients of the loss function with respect to the parameters of neural networks. The gradients are computed layer by layer, starting from the output layer and moving backward through the network.

Regularization is a technique used to prevent overfitting in machine learning models. In NLP, L1 and L2 regularization are commonly used to add a penalty term to the loss function, which encourages the model to have smaller parameter values. The gradients of the penalty term are then added to the gradients of the loss function when updating the parameters.